The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc $\mathbb D^n.$ In the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. An alternative approach to the Carath\'{e}odory-Fej\'{e}r interpolation problem, in the special case of $n=2,$ adapting a theorem of Kor\'{a}nyi and Puk\'{a}nzsky is given. As a consequence, a class of polynomials are isolated for which a complete solution to the Carath\'{e}odory-Fej\'{e}r interpolation problem is easily obtained. A natural generalization of the Hankel operators on the Hardy space of $H^2(\mathbb T^2)$ then becomes apparent. Many of our results remain valid for any $n\in \mathbb N,$ however, the computations are somewhat cumbersome for $n>2$ and are omitted. The inequality $\lim_{n\to \infty}C_2(n)\leq 2 K^\mathbb C_G$, where $K_G^\mathbb C$ is the complex Grothendieck constant and $$C_2(n)=\sup\big\{\|p(\boldsymbol T)\|:\|p\|_{\mathbb D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1 \big\}$$ is due to Varopoulos. Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples $\boldsymbol T:=(T_1,\ldots,T_n)$ of contractions. We show that $$\lim_{n\to \infty}C_2(n)\leq \frac{3\sqrt{3}}{4} K^\mathbb C_G$$ obtaining a slight improvement in the inequality of Varopoulos. We show that the normed linear space $\ell^1(n),$ $n>1,$ has no isometric embedding into $k\times k$ complex matrices for any $k\in \mathbb N$ and discuss several infinite dimensional operator space structures on it.